Nonlinear Income Effects in Random Utility Models: Revisiting the ...

Nonlinear Income Effects in Random Utility Models: Revisiting the Accuracy of the Representative Consumer Approximation March 16, 2009

Constant I. Tra CBER and Department of Economics University of Nevada, Las Vegas

Contact: Constant I. Tra, 4505 S. Maryland Parkway, Box 456002, Las Vegas, NV 89154-6002, USA Telephone: 702-895-3191 Fax: 702-895-3606 [email protected]

Acknowledgments: I am grateful to Ted McConnell for valuable comments on earlier versions on this paper. I thank Rennae Daneshvary for her assistance in editing this paper.

Nonlinear Income Effects in Random Utility Models: Revisiting the Accuracy of the Representative Consumer Approximation Abstract

The paper investigates the implications of nonlinear income effects in random utility models (RUM) for measuring non-marginal welfare impacts. A popular approach in applied welfare analysis is to approximate the expected compensating variation (cv) for an amenity change as the cv of a representative consumer whose indirect utility is given by the expected maximum utility. However, this approach can be misleading in the case of non-marginal changes as it implies that changes in income do not affect the consumer’s choice. In this case the true expected cv can be obtained via simulation. Empirical applications to recreational demand find that the bias from the representative approach is small. This paper re-evaluates the accuracy of the representative consumer approximation in the context of measuring the general equilibrium welfare impacts of large environmental changes. Our findings suggest that, though the representative consumer approximation could lead to biased point estimates of the expected cv, this bias is overwhelmed by the size of the confidence intervals that result from the empirical estimation of household preferences.

JEL Classification: Q51, R21 Keywords: compensating variation, nonlinear income effects, discrete choice

1

Introduction

Random utility models (RUM) are often used to evaluate compensating variation (cv) measures for changes in environmental amenities. In the RUM framework, the cv is itself a random variable because the indirect utility function contains a stochastic error term. Hence, it is common practice in applied work to use the expected value of the cv welfare measure. Recovering the expected cv, however, poses a computational challenge, as it does not have a general closed form expression (see for e.g., McFadden, 1999). The vast majority of the applications of RUM models have relied on the assumption that the marginal utility of income is constant across alternatives in order to recover the expected cv. This assumption greatly simplifies the computation of the expected cv. This is because the maximum utility is a linear function of income and, hence, a closed form expression for both the cv and its expected value can be easily computed. The constant marginal utility of income assumption can, however, be restrictive in some applications as it implies that changes in income do not affect the consumer’s choice. In the RUM framework the presence of an income effect implies that a consumer who is facing the same set of alternatives and alternative attributes may make different choices at different levels of his/her real income (Bockstael and McConnell, 2007). This income effect will be negligible when alternatives do not differ significantly in cost. A typical example would be the choice among local recreational beaches which are all available for a day trip (see e.g. Bockstael and McConnell, 2007). Many recreation demand applications of the RUM framework tend to fit into this scenario. There, however, are situations when the cost of choice alternatives can vary significantly. The choice of residential housing is such a case. In regional housing markets, housing prices can vary significantly across different neighborhoods. As a result, we

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can expect changes in real income to have a significant impact on household location choices, which means that income effects can no longer be assumed linear. The computation of the expected cv then becomes a real issue. Two approaches have been suggested for calculating the expected cv in the presence of nonlinear income effects. The first approach is to approximate the expected cv as the income compensation which equates the expected maximum utility of a representative consumer in the before and after scenarios.1,2 The resulting estimate of the expected cv has been referred to as the representative consumer approximation. From a computational cost standpoint this approach is very appealing as it requires only computing the solution to an implicit function for each consumer. McFadden (1999) shows analytically that the representative consumer approximation of the expected cv can produce biased welfare estimates when large amenity changes are considered. Alternatively, he proposed a general simulation approach for recovering the exact expected cv measure in RUMs with nonlinear income effects.3 The main limitation of the simulation approach is the computational intensity. Herriges and Kling (1999) provided the first empirical investigation of the extent to which the representative consumer approach correctly approximates the true expected cv. In their study of fishing-mode choices by California anglers, the authors find that the point estimates of the expected cv are virtually identical for the representative consumer approach and McFadden’s exact welfare computation, which suggests that the representative consumer approach indeed provides an accurate approximation of the expected cv. However, the authors also find no 1

Note that the true expected cv is defined as the expected value of the income compensation that equates the maximum utility before and after an amenity change. 2 This approach was originally suggested by Hanneman (1985). The first empirical use was by Bockstael, Hanneman and Kling (1987). A later application by Morey et al. (1993) is often cited for this approach. 3 Recently, Dagsvik and Karlström (2007) have suggested a general closed form expression for the expected cv in nonlinear RUM models using a random expenditure function. This closed form expression also involves the computation of a numerical integral.

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significant difference in welfare effects between the nonlinear income specification and the linear income specification, which suggests that income effects are small. Hence, the Herriges and Kling (1999) study may not provide an appropriate setting for investigating the empirical bias that arises in the representative consumer approximation of the expected cv in the presence of income effects. This study investigates the implication of the representative consumer approximation to the expected cv for measuring non-marginal welfare impacts in RUMs when income effects are nonlinear. The empirical investigation evaluates the general equilibrium benefits of the reductions in ozone levels that occurred across the Los Angeles area between 1990 and 2000. The Los Angeles area housing market provides an ideal setting for the presence of nonlinear income effects. Housing prices varied significantly across the study area in 1990. The neighborhood mean rental price of housing ranged from $580 to $1,230. In addition, average ozone concentrations across the study area fell by nearly 40 percent between 1990 and 2000.4 All these facts would suggest that income effects are likely to be significant. Using point estimates and confidence bounds, we compare and contrast the cv measures obtained using both the representative consumer approach and McFadden’s exact welfare computation. As predicted by McFadden (1999), we find that the point estimates from the representative consumer E(cv) approximation are quite different from the exact E(cv) estimates. Also, as suggested by Herriges and Kling (1999), we find that the size of the confidence intervals that result from the empirical estimation of household preferences is large enough to outweigh this bias. The remainder of the paper is outlined as follows. Section 2 describes the derivation of the welfare measures. Section 3 describes the empirical estimation and the data. Section 4 discusses the welfare results, and section 5 concludes. 4

See Tra (2007) for further discussion of ozone changes in the Los Angeles area.

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2 2.1

The empirical model Model specification

The empirical application estimates an equilibrium model of housing choice for the Los Angeles metropolitan area.5 Households chose their residential location j from a discrete set of 5,000 housing products (H) in 1990.6 The utility that a household i derives from a residential location j is given by:

Vij = α1log(y − pj)+ α2q j + α 3q jlog(y − pj)+ Xjβ + ξj + εij ,

(1)

where y represents the household’s monthly income, ph is the monthly rental price of the housing product j. qj is the ozone concentration at the residential location j and Xj is a vector of housing and neighborhood attributes. α1, α2, α3 and β are the preference parameters that will be estimated. ξj is a house-specific error term that is used to capture unobserved characteristics of the housing product j. εij is a random error term with a type I extreme value distribution.7 2.2

Data

We estimate the parameters of households’ preferences from a cross section of microdata that includes household characteristics, housing characteristics, neighborhood air quality and other neighborhood variables. Households and housing characteristics are obtained from the 1990 and 2000 Census Public Use Microdata 5-percent Sample (PUMS). A complete description of the data sources and data issues is provided in Tra (2007).

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A more detailed description of this model and its estimation are provided in Tra (2007). A housing product is defined as a collection of houses with identical observed characteristics and located within the same neighborhood. 7 This assumption gives rise to the multinomial logit (MNL) model. Herriges and Kling (1999) find that the exact and representative consumer welfare measures do not differ significantly for the MNL and Charter fishing nesting structure. Hence, we do not attempt to investigate alternative nesting structures. 6

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2.3

Estimating the preference parameters

The parameters (α1, α2, α3 and β) of the household indirect utility function in equation (1) are estimated from a multinomial logit model. The estimation follows the two-stage approach proposed by Bayer et al. (2005). In the first stage we estimate, via maximum likelihood, (J-1) alternative-specific constants8 (δh) and the parameters (α1 and α3) characterizing the householdspecific tastes. The second stage estimates the vector of mean taste parameters (α2 and β) via least-squares regression using the estimated vector of alternative constants as the dependent variable. Table 1 summarizes the results of the estimation. Model 1 estimates the benchmark specification which is used in the welfare estimation. The other models provide robustness checks. The estimated ozone coefficients imply a marginal willingness to pay (MWTP) of $92 for a 1 percent reduction in the 1990 average ozone concentration. This estimate is consistent with the existing empirical literature.9 3

Recovering the general equilibrium welfare measures10

For a policy regime which leads to air quality changes from q0 to q1, and housing price changes from p0 to p1, the cv for is implicitly defined by:

Max{v h(y − p0j,q 0j,εj)} = Max{v k(y − p1k − cv ,q 1k,εk)}. j∈Η

k∈Η

(2)

Alternatively, we can characterize the general equilibrium welfare impact of a policy change in terms of the household’s equivalent variation (ev). The ev for the policy change is implicitly defined as:

Max{v j(y − p0j + ev ,q 0j,εj)} = Max{v k(y − p1k,q 1k,εk )}. j∈Η

k∈Η

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Note: The Hth alternative constant is set to zero. Estimates of the MWTP for air quality range from $18 to $181 in the literature (Sieg et al., 2004). 10 Hanneman (1985) and McFadden (1999) provide a detailed theoretical treatment of the welfare measurement issues in RUM models. 9

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(3)

In the absence of income effects, the cv is equal to the ev. Hence, an indirect way to test for nonlinear income effects would be to compare the cv and ev estimates for a given amenity change. The welfare measures defined by equations (2) and (3) are random variables. This is directly due to the stochastic error term ε present in the household utility. The remainder of this section discusses alternative numerical approaches for evaluating the expected values for the cv and ev welfare measures. 3.1

Recovering the exact expected cv measure

McFadden (1999) suggested a general simulation approach for recovering the exact expected cv measure.11 For each household, T random draws of the stochastic error ε = (ε1,…, εH)’ are obtained. For every draw t the household’s cv measure, as defined by equation (2), is computed via a simple one-dimensional search algorithm. To implement the simulation procedure we generate T = 100 independent vectors of pseudorandom extreme value variables. The size of T was selected on the basis of a Monte Carlo experiment as suggested by Herriges and Kling (1999). The estimation of Eˆ (cv) using T iterations was repeated 100 times. We found that after T = 50, the estimated mean cv was very similar over the 100 trials. The standard deviation was roughly 4 percent of the mean value across the 100 trials for T = 50. By T = 100 the standard deviation was reduced to roughly 1 percent of the mean compensating variation across the 100 trials. The estimate of the household’s expected cv, denoted Eˆ (cv), is obtained as the average of the cv measures across the T draws:

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Dagsvik and Karlström (2005) show that a closed form expression for the expected cv measure can be defined in terms of the random expenditure function. However, the evaluation of the expected cv also involves the computation of a numerical integral.

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T

1 Eˆ (cv)= ∑ cvt . T t =1

(4)

McFadden (1997) has shown that Eˆ (cv) converges almost surely to E(cv) as T→∞. The complete simulation procedure is outlined by McFadden (1999) for the general case of an unobserved taste error term with a generalized extreme value (GEV) distribution.

3.2

A representative consumer approximation of the expected cv

An alternative, simpler approach is to approximate the expected cv by the income compensation which equates the expected maximum utility of a representative consumer before and after the air quality changes. This approximation of E(cv) is defined implicitly as:

[

]

EMax{v j(y − p0j,q 0j,εj)} = E Max{v k(y − p1k − cv ,q 1k,εk )} .  j∈Η  k∈Η

(5)

When the stochastic error ε is additive and ε is drawn from an extreme value (ev) distribution, the expression for the expected maximum utility is given by:

E[Vj(y − pj,q j,εj)]= log∑ Exp[v j(y − pj,q j)] + c

(6)

j∈Η

Where, c is a mathematical constant which cancels out as it appears on both sides of equation ~ (5). The computation of E(cv) is achieved via a one-dimensional search algorithm.

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Results

Table 2 summarizes the welfare results. The first column reports the mean welfare measures calculated from the point estimates of the household preference parameters. The point estimate of the exact E(cv) is $3,017 for the reductions in ozone levels that occurred in the Los Angeles area between 1990 and 2000. The mean point estimate for the E(cv) approximation is 8 percent larger ($3,262).

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Standard errors and confidence intervals for the point estimates of E(cv) are obtained via simulation. We take 500 normal random draws of the vector of preference parameters, using the point estimates and asymptotic covariance obtained from the maximum likelihood estimation in Tra (2007).12 For each parameter draw, the E(cv) measure is obtained. This provides an empirical distribution of the mean E(cv) estimate. Columns 2 through 8 of Table 2 report summary statistics of the empirical distribution of the E(cv) estimate. We find that the difference between the exact and approximation measures of E(cv) is swamped by the size of the confidence intervals for each welfare measure. This point can be easily observed from Figure 1. The asterisk inside the bar graph indicates the point estimate of the welfare measure, whereas the shaded area shows the 95 percent confidence region around the estimate. These results suggest that, though the representative consumer approximation could lead to biased point estimates of E(cv), this bias is overwhelmed by the size of the confidence intervals that result from the empirical estimation of household preferences. The bottom two rows of Table 2 report the point estimate and empirical distribution of the expected equivalent variation, E(ev). The relationship between the exact and approximation measures of E(ev) is the same as for the case of E(cv). Theoretically, E(ev) should be equal to E(cv) when there are no income effects. We find that the point estimate of E(ev) is about 16 percent larger than that of E(cv) for both the exact and approximation measures. However, these differences are small relative to the confidence intervals for the welfare estimates. This would suggest that the difference in welfare estimates that would arise from a model which assumes

12

The number of draws was restricted because of computational time. The computation of the E(cv) for the 500 draws required approximately 20 days using a Redhat Linux server from the University of Maryland’s GRACE cluster.

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linear income effects would not be statistically significant.13 This result is also consistent with the finding that the exact and approximation measures of E(cv) are not significantly different.

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Conclusion

This paper has investigated the performance of the representative consumer approach to approximating the expected cv in nonlinear RUMs. Our findings provide some support for both the analytical intuition provided by McFadden (1999) and the empirical results from Herriges and Kling (1999). As predicted by McFadden (1999), we find that the point estimates from the representative consumer E(cv) approximation are quite different from the exact E(cv) estimates. Also, as suggested by Herriges and Kling (1999), we find that the size of the confidence intervals that result from the empirical estimation of household preferences is large enough to outweigh this bias. Our findings suggest that, though the representative consumer approximation could lead to biased point estimates of the E(cv), this bias is overwhelmed by the size of the confidence intervals that result from the empirical estimation of household preferences.

13

That E(ev) does not differ significantly from E(cv) could also be a matter of poor income data. Household income is self reported in the U.S. Census, which could lead to considerable noise in the income variable.

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References Bayer, P., R. McMillan and K. Rueben. 2005. “An Equilibrium Model of Sorting in an Urban Market.” NBER Working Paper: 10865. Bockstael, N., M. Hanneman and C. Kling. 1987 “Estimating the Value of Water Quality Improvements in a Recreational Demand Framework.” Water Resources Research 23(5): 273-302. Bockstael, N.E. and K.E. McConnell. Environmental Valuation with Revealed Preferences: A Theoretical Guide to Empirical Models. Kluwer Publishing (forthcoming) Dagsvik, J. K. and A. Karlström. 2007. “Compensating Variation and Hicksian Choice Probabilities in Random Utility Models That Are Nonlinear in Income.” The Review of Economic Studies 72(1): 57-76. Hanneman, W. M. 1985. “Welfare Analysis with Discrete Choice Models.” University of California, Agricultural and Resource Economics, Giannini Foundation Working Paper. Herriges, J. and C. Kling. 1999. “Nonlinear Income Effects in Random Utility Models.” The Review of Economics and Statistics 81(1): 62-72. McFadden, D. 1999. “Computing Willingness-to-Pay in Random Utility Models.” in J. Moore, R. Riezman and J. Melvin, eds., Trade, Theory and Econometrics: Essays in Honor of John S. Chipman, London: Routledge. Morey, E. 1999. “Two RUMs Uncloaked: Nested-Logit Models of Participation and Site Choice.” in J. Herriges and C. Kling, eds., Valuing Recreation and the Environment: Revealed Preference Methods in Theory and Practice, Cheltenham, U.K. and Northampton, Mass.: Elgar. Morey, E., R. D. Rowe and M. Watson. 1993. “A Repeated Nested-Logit Model of Atlantic Salmon Fishing.” The American Journal of Agricultural Economics 75(3): 578-92. Sieg, H., V. K. Smith, H. S. Banzhaf and R. Walsh. 2004. “Estimating the General Equilibrium Benefits of Large Changes in Spatially Delineated Public Goods.” International Economic Review 45(4): 1047-77. Tra, C. 2007. Evaluating the Equilibrium Welfare Impacts of the 1990 Clean Air Act Amendments in the Los Angeles Area. Ph.D. dissertation, Department of Agricultural and Resource Economics, University of Maryland, College Park, MD. URL: http://www.lib.umd.edu/drum/bitstream/1903/7236/1/umi-umd-4635.pdf.

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Table 1: Estimation Results

Model 1‡

Model 2

Model 3

First Stage MLE Log(y-p) Ozone * Log(y-p) Bedrooms * Household size Single family * Children under 18 Math * College educated head Log FBI crime index * Log(y-p) Within household’s employment zone

1.554** -0.30** 0.052** 0.309** 0.037** -0.003** 2.382**

1.672** -0.29** 0.053** 0.308** 0.029** -0.017** -

1.520** -0.26** 0.055** 0.270** 0.043** -0.003** 2.466**

Log-Likelihood McFadden pseudo-R2 Observations

-93,963 0.303 44,291

-100,905 -106,284 0.252 0.275 44,291 48,139

Second Stage OLS † Intercept Year = 2000 Orange County Riverside County San Bernardino County Bedrooms Single-family dwelling Owned Math test score Log FBI crime index Log elevation PUMA is on Pacific coastline Log density Ozone

-3.010** 1.074** 0.416** -0.015 -0.059 0.030** 0.621** -0.031 0.009** 0.085** 0.094** 0.416** 0.060** 1.04

-2.406** 1.070** 0.315** -0.072 -0.165** 0.032** 0.624** -0.038 0.005 0.082** 0.079** 0.461** 0.016 1.40

-2.992** 0.254** 0.276** -0.152** -0.164** 0.054** 0.457** 0.101** 0.01** 0.112** 0.089** 0.397** 0.080** 0.36

R2 Observations

0.175 10,612

0.155 10,612

0.094 18,713

Notes: ** Significant at 1% level. * Significant at 5% level. † Standard errors are computed using White’s robust covariance matrix. ‡ Model 1: Benchmark specification used in the simulation and welfare analysis. Model 2: Estimates the first stage without the “employment” variable. Model 3: Characterizes residential locations using a more refined product space.

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Table 2: Welfare Impacts of Air Quality Improvements in the Los Angeles Area (1990-2000) †

Welfare Measure

Calculated from Parameter Mean

Mean Welfare from 500 Draws

E(cv) exact E(cv) approx.

3,017 3,502

E(ev) exact E(ev) approx.

3,262 3,844

Lower Tail 2.5%

5%

3,017 3,504

2,590 2,963

2,633 3,034

3,265 3,850

2,766 3,202

2,816 3,285

Median Welfare from 500 Draws

Upper Tail

Standard Deviation from 500 Draws

95%

97.5%

3,021 3,511

3,381 3,980

3,465 4,048

216 273

3,271 3,859

3,702 4,433

3,794 4,519

256 331

Note: † cv: Compensating variation. ev: Equivalent variation. All monetary values are in annual 1990 $.

Figure 1: Point estimates and Confidence Bounds for Welfare Measures

E(cv) Exact

*

E(cv) Approx.

*

E(ev) Exact

*

E(ev) Approx.

2,500

*

3,000

3,500

4,000

12

4,500

5,000